LMFDB - Embedded newform 48.27.e.b.17.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,27,Mod(17,48)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 27, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("48.17");

S:= CuspForms(chi, 27);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 27 $
Character orbit:	$[\chi]$	$=$	48.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$205.580601950$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 11609466 x^{6} - 3416571600 x^{5} + 38618090622117 x^{4} + \cdots + 60\!\cdots\!96 $ x^8 - 11609466x^6 - 3416571600x^5 + 38618090622117x^4 + 19832313245955600x^3 - 25659229618759655524x^2 - 8410094250198512820000x + 6059395276689883547331396
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{65}\cdot 3^{36}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 6)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			17.2
Root			$-2107.01 + 1.41421i$ of defining polynomial
Character	$\chi$	$=$	48.17
Dual form			48.27.e.b.17.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.54011e6 + 412209. i) q^{3} +1.23273e9i q^{5} +6.52628e10 q^{7} +(2.20203e12 - 1.26970e12i) q^{9} +O(q^{10})$	\(q+(-1.54011e6 + 412209. i) q^{3} +1.23273e9i q^{5} +6.52628e10 q^{7} +(2.20203e12 - 1.26970e12i) q^{9} +4.72129e13i q^{11} +3.70280e14 q^{13} +(-5.08144e14 - 1.89855e15i) q^{15} -2.98298e13i q^{17} -7.01278e16 q^{19} +(-1.00512e17 + 2.69019e16i) q^{21} +4.09515e17i q^{23} -2.95138e16 q^{25} +(-2.86800e18 + 2.86318e18i) q^{27} -1.48710e19i q^{29} -9.71798e18 q^{31} +(-1.94616e19 - 7.27132e19i) q^{33} +8.04516e19i q^{35} -2.55385e19 q^{37} +(-5.70273e20 + 1.52633e20i) q^{39} +1.47371e21i q^{41} +1.20352e21 q^{43} +(1.56520e21 + 2.71452e21i) q^{45} -2.01672e20i q^{47} -5.12824e21 q^{49} +(1.22961e19 + 4.59413e19i) q^{51} +1.74636e22i q^{53} -5.82009e22 q^{55} +(1.08005e23 - 2.89073e22i) q^{57} +1.75317e23i q^{59} +8.70380e22 q^{61} +(1.43711e23 - 8.28641e22i) q^{63} +4.56456e23i q^{65} -4.13304e23 q^{67} +(-1.68806e23 - 6.30700e23i) q^{69} -1.95086e24i q^{71} +2.56630e24 q^{73} +(4.54547e22 - 1.21659e22i) q^{75} +3.08125e24i q^{77} +7.16146e24 q^{79} +(3.23682e24 - 5.59183e24i) q^{81} +1.27711e25i q^{83} +3.67722e22 q^{85} +(6.12996e24 + 2.29030e25i) q^{87} +2.12054e25i q^{89} +2.41655e25 q^{91} +(1.49668e25 - 4.00584e24i) q^{93} -8.64489e25i q^{95} -6.17710e25 q^{97} +(5.99461e25 + 1.03964e26i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2194920 q^{3} - 183211263760 q^{7} - 4853931303096 q^{9}+O(q^{10}) $ 8 * q - 2194920 * q^3 - 183211263760 * q^7 - 4853931303096 * q^9	$ 8 q - 2194920 q^{3} - 183211263760 q^{7} - 4853931303096 q^{9} + 512429022548560 q^{13} + 10\!\cdots\!40 q^{15}+ \cdots - 49\!\cdots\!36 q^{99}+O(q^{100}) $ 8 * q - 2194920 * q^3 - 183211263760 * q^7 - 4853931303096 * q^9 + 512429022548560 * q^13 + 1064426665559040 * q^15 - 13250502825732304 * q^19 - 376087240418977584 * q^21 - 5623831579932691000 * q^25 - 11144542948278241320 * q^27 + 31284409294003569776 * q^31 - 27135100230528798720 * q^33 - 884494012218278308400 * q^37 - 1550610890749922311440 * q^39 + 5161357724259210085040 * q^43 + 20490779480380306513920 * q^45 - 29984991896355791854440 * q^49 - 55593505357359326429184 * q^51 + 76225105088113564293120 * q^55 + 66535314651975639764880 * q^57 + 235221054408754640644432 * q^61 + 635461358798759743298160 * q^63 - 4218736573110688595120080 * q^67 - 2608639506957020943089664 * q^69 + 1648536069669962768198800 * q^73 - 2584159434345978668025000 * q^75 + 9378866636279258011040624 * q^79 - 746539638785265647226744 * q^81 - 41457721476781794937282560 * q^85 - 55271553720839002987729920 * q^87 + 71633324581325574226388320 * q^91 + 18549099857564620426167120 * q^93 - 92292347027073928712964080 * q^97 - 4908019641378975318951936 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/48\mathbb{Z}\right)^\times$.

$n$	$17$	$31$	$37$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−1.54011e6	+	412209.i	−0.965998	+	0.258548i
$3$	−1.54011e6	+	412209.i	−0.965998	+	0.258548i
$4$		0			0
$5$			1.23273e9i			1.00985i	0.863162	+	0.504927i	$0.168481\pi$
$5$			1.23273e9i			1.00985i	−0.863162	+	0.504927i	$0.831519\pi$
$6$		0			0
$7$	6.52628e10			0.673583			0.336792	−	0.941579i	$-0.390658\pi$
$7$	6.52628e10			0.673583			0.336792	+	0.941579i	$0.390658\pi$
$8$		0			0
$9$	2.20203e12	−	1.26970e12i	0.866306	−	0.499514i
$10$		0			0
$11$			4.72129e13i			1.36759i	0.729675	+	0.683794i	$0.239671\pi$
$11$			4.72129e13i			1.36759i	−0.729675	+	0.683794i	$0.760329\pi$
$12$		0			0
$13$	3.70280e14			1.22255			0.611274	−	0.791419i	$-0.290657\pi$
$13$	3.70280e14			1.22255			0.611274	+	0.791419i	$0.290657\pi$
$14$		0			0
$15$	−5.08144e14	−	1.89855e15i	−0.261096	−	0.975518i
$16$		0			0
$17$		−	2.98298e13i		−	0.00301172i	−0.999999	−	0.00150586i	$-0.999521\pi$
$17$		−	2.98298e13i		−	0.00301172i	0.999999	−	0.00150586i	$-0.000479331\pi$
$18$		0			0
$19$	−7.01278e16			−1.66761			−0.833803	−	0.552062i	$-0.813841\pi$
$19$	−7.01278e16			−1.66761			−0.833803	+	0.552062i	$0.813841\pi$
$20$		0			0
$21$	−1.00512e17	+	2.69019e16i	−0.650681	+	0.174154i
$22$		0			0
$23$			4.09515e17i			0.812472i	0.913768	+	0.406236i	$0.133159\pi$
$23$			4.09515e17i			0.812472i	−0.913768	+	0.406236i	$0.866841\pi$
$24$		0			0
$25$	−2.95138e16			−0.0198064
$26$		0			0
$27$	−2.86800e18	+	2.86318e18i	−0.707702	+	0.706512i
$28$		0			0
$29$		−	1.48710e19i		−	1.44933i	−0.689103	−	0.724663i	$-0.741995\pi$
$29$		−	1.48710e19i		−	1.44933i	0.689103	−	0.724663i	$-0.258005\pi$
$30$		0			0
$31$	−9.71798e18			−0.397992			−0.198996	−	0.980000i	$-0.563768\pi$
$31$	−9.71798e18			−0.397992			−0.198996	+	0.980000i	$0.563768\pi$
$32$		0			0
$33$	−1.94616e19	−	7.27132e19i	−0.353588	−	1.32109i
$34$		0			0
$35$			8.04516e19i			0.680221i
$36$		0			0
$37$	−2.55385e19			−0.104851			−0.0524256	−	0.998625i	$-0.516695\pi$
$37$	−2.55385e19			−0.104851			−0.0524256	+	0.998625i	$0.516695\pi$
$38$		0			0
$39$	−5.70273e20	+	1.52633e20i	−1.18098	+	0.316088i
$40$		0			0
$41$			1.47371e21i			1.59303i	0.604621	+	0.796513i	$0.293325\pi$
$41$			1.47371e21i			1.59303i	−0.604621	+	0.796513i	$0.706675\pi$
$42$		0			0
$43$	1.20352e21			0.700428			0.350214	−	0.936670i	$-0.386109\pi$
$43$	1.20352e21			0.700428			0.350214	+	0.936670i	$0.386109\pi$
$44$		0			0
$45$	1.56520e21	+	2.71452e21i	0.504437	+	0.874843i
$46$		0			0
$47$		−	2.01672e20i		−	0.0369295i	−0.999830	−	0.0184647i	$-0.994122\pi$
$47$		−	2.01672e20i		−	0.0369295i	0.999830	−	0.0184647i	$-0.00587784\pi$
$48$		0			0
$49$	−5.12824e21			−0.546285
$50$		0			0
$51$	1.22961e19	+	4.59413e19i	0.000778675	+	0.00290932i
$52$		0			0
$53$			1.74636e22i			0.670731i	0.942088	+	0.335365i	$0.108860\pi$
$53$			1.74636e22i			0.670731i	−0.942088	+	0.335365i	$0.891140\pi$
$54$		0			0
$55$	−5.82009e22			−1.38107
$56$		0			0
$57$	1.08005e23	−	2.89073e22i	1.61091	−	0.431157i
$58$		0			0
$59$			1.75317e23i			1.67012i	0.550159	+	0.835060i	$0.314567\pi$
$59$			1.75317e23i			1.67012i	−0.550159	+	0.835060i	$0.685433\pi$
$60$		0			0
$61$	8.70380e22			0.537553			0.268776	−	0.963203i	$-0.413381\pi$
$61$	8.70380e22			0.537553			0.268776	+	0.963203i	$0.413381\pi$
$62$		0			0
$63$	1.43711e23	−	8.28641e22i	0.583529	−	0.336464i
$64$		0			0
$65$			4.56456e23i			1.23460i
$66$		0			0
$67$	−4.13304e23			−0.753870			−0.376935	−	0.926240i	$-0.623022\pi$
$67$	−4.13304e23			−0.753870			−0.376935	+	0.926240i	$0.623022\pi$
$68$		0			0
$69$	−1.68806e23	−	6.30700e23i	−0.210063	−	0.784846i
$70$		0			0
$71$		−	1.95086e24i		−	1.67443i	−0.546875	−	0.837214i	$-0.684183\pi$
$71$		−	1.95086e24i		−	1.67443i	0.546875	−	0.837214i	$-0.315817\pi$
$72$		0			0
$73$	2.56630e24			1.53501			0.767503	−	0.641046i	$-0.221499\pi$
$73$	2.56630e24			1.53501			0.767503	+	0.641046i	$0.221499\pi$
$74$		0			0
$75$	4.54547e22	−	1.21659e22i	0.0191330	−	0.00512091i
$76$		0			0
$77$			3.08125e24i			0.921185i
$78$		0			0
$79$	7.16146e24			1.53408			0.767042	−	0.641597i	$-0.221728\pi$
$79$	7.16146e24			1.53408			0.767042	+	0.641597i	$0.221728\pi$
$80$		0			0
$81$	3.23682e24	−	5.59183e24i	0.500971	−	0.865464i
$82$		0			0
$83$			1.27711e25i			1.43951i	0.694229	+	0.719755i	$0.255746\pi$
$83$			1.27711e25i			1.43951i	−0.694229	+	0.719755i	$0.744254\pi$
$84$		0			0
$85$	3.67722e22			0.00304140
$86$		0			0
$87$	6.12996e24	+	2.29030e25i	0.374721	+	1.40005i
$88$		0			0
$89$			2.12054e25i			0.964666i	0.875988	+	0.482333i	$0.160210\pi$
$89$			2.12054e25i			0.964666i	−0.875988	+	0.482333i	$0.839790\pi$
$90$		0			0
$91$	2.41655e25			0.823489
$92$		0			0
$93$	1.49668e25	−	4.00584e24i	0.384460	−	0.102900i
$94$		0			0
$95$		−	8.64489e25i		−	1.68404i
$96$		0			0
$97$	−6.17710e25			−0.917809			−0.458904	−	0.888486i	$-0.651758\pi$
$97$	−6.17710e25			−0.917809			−0.458904	+	0.888486i	$0.651758\pi$
$98$		0			0
$99$	5.99461e25	+	1.03964e26i	0.683130	+	1.18475i
$100$		0			0
$101$			1.41146e26i			1.24020i	0.784524	+	0.620099i	$0.212908\pi$
$101$			1.41146e26i			1.24020i	−0.784524	+	0.620099i	$0.787092\pi$
$102$		0			0
$103$	−1.15485e26			−0.786399			−0.393199	−	0.919453i	$-0.628632\pi$
$103$	−1.15485e26			−0.786399			−0.393199	+	0.919453i	$0.628632\pi$
$104$		0			0
$105$	−3.31629e25	−	1.23905e26i	−0.175870	−	0.657093i
$106$		0			0
$107$		−	7.52135e25i		−	0.312109i	−0.987748	−	0.156055i	$-0.950122\pi$
$107$		−	7.52135e25i		−	0.312109i	0.987748	−	0.156055i	$-0.0498776\pi$
$108$		0			0
$109$	1.76550e26			0.575868			0.287934	−	0.957650i	$-0.407032\pi$
$109$	1.76550e26			0.575868			0.287934	+	0.957650i	$0.407032\pi$
$110$		0			0
$111$	3.93322e25	−	1.05272e25i	0.101286	−	0.0271091i
$112$		0			0
$113$		−	5.42487e26i		−	1.10757i	−0.832661	−	0.553783i	$-0.813184\pi$
$113$		−	5.42487e26i		−	1.10757i	0.832661	−	0.553783i	$-0.186816\pi$
$114$		0			0
$115$	−5.04823e26			−0.820478
$116$		0			0
$117$	8.15368e26	−	4.70143e26i	1.05910	−	0.610680i
$118$		0			0
$119$		−	1.94678e24i		−	0.00202865i
$120$		0			0
$121$	−1.03724e27			−0.870299
$122$		0			0
$123$	−6.07478e26	−	2.26969e27i	−0.411874	−	1.53886i
$124$		0			0
$125$			1.80053e27i			0.989853i
$126$		0			0
$127$	−1.64192e27			−0.734352			−0.367176	−	0.930152i	$-0.619675\pi$
$127$	−1.64192e27			−0.734352			−0.367176	+	0.930152i	$0.619675\pi$
$128$		0			0
$129$	−1.85356e27	+	4.96102e26i	−0.676612	+	0.181094i
$130$		0			0
$131$			3.57069e27i			1.06715i	0.845753	+	0.533575i	$0.179152\pi$
$131$			3.57069e27i			1.06715i	−0.845753	+	0.533575i	$0.820848\pi$
$132$		0			0
$133$	−4.57674e27			−1.12327
$134$		0			0
$135$	−3.52953e27	−	3.53548e27i	−0.713474	−	0.714676i
$136$		0			0
$137$		−	2.69983e27i		−	0.450785i	−0.974268	−	0.225393i	$-0.927634\pi$
$137$		−	2.69983e27i		−	0.450785i	0.974268	−	0.225393i	$-0.0723665\pi$
$138$		0			0
$139$	1.15296e28			1.59449			0.797244	−	0.603658i	$-0.206291\pi$
$139$	1.15296e28			1.59449			0.797244	+	0.603658i	$0.206291\pi$
$140$		0			0
$141$	8.31310e25	+	3.10598e26i	0.00954805	+	0.0356738i
$142$		0			0
$143$			1.74820e28i			1.67194i
$144$		0			0
$145$	1.83320e28			1.46361
$146$		0			0
$147$	7.89808e27	−	2.11391e27i	0.527711	−	0.141241i
$148$		0			0
$149$		−	1.77789e27i		−	0.0996516i	−0.998758	−	0.0498258i	$-0.984133\pi$
$149$		−	1.77789e27i		−	0.0996516i	0.998758	−	0.0498258i	$-0.0158666\pi$
$150$		0			0
$151$	2.36660e26			0.0111539			0.00557693	−	0.999984i	$-0.498225\pi$
$151$	2.36660e26			0.0111539			0.00557693	+	0.999984i	$0.498225\pi$
$152$		0			0
$153$	−3.78749e25	−	6.56863e25i	−0.00150440	−	0.00260907i
$154$		0			0
$155$		−	1.19797e28i		−	0.401914i
$156$		0			0
$157$	−4.48717e28			−1.27431			−0.637157	−	0.770734i	$-0.719890\pi$
$157$	−4.48717e28			−1.27431			−0.637157	+	0.770734i	$0.719890\pi$
$158$		0			0
$159$	−7.19867e27	−	2.68960e28i	−0.173416	−	0.647925i
$160$		0			0
$161$			2.67261e28i			0.547267i
$162$		0			0
$163$	2.42227e28			0.422458			0.211229	−	0.977437i	$-0.432253\pi$
$163$	2.42227e28			0.422458			0.211229	+	0.977437i	$0.432253\pi$
$164$		0			0
$165$	8.96359e28	−	2.39909e28i	1.33411	−	0.357072i
$166$		0			0
$167$		−	6.59270e28i		−	0.838975i	−0.907761	−	0.419488i	$-0.862210\pi$
$167$		−	6.59270e28i		−	0.838975i	0.907761	−	0.419488i	$-0.137790\pi$
$168$		0			0
$169$	4.53737e28			0.494626
$170$		0			0
$171$	−1.54424e29	+	8.90412e28i	−1.44466	+	0.832993i
$172$		0			0
$173$			1.62424e29i			1.30633i	0.757218	+	0.653163i	$0.226558\pi$
$173$			1.62424e29i			1.30633i	−0.757218	+	0.653163i	$0.773442\pi$
$174$		0			0
$175$	−1.92616e27			−0.0133413
$176$		0			0
$177$	−7.22672e28	−	2.70008e29i	−0.431806	−	1.61333i
$178$		0			0
$179$			1.69124e29i			0.873206i	0.899654	+	0.436603i	$0.143819\pi$
$179$			1.69124e29i			0.873206i	−0.899654	+	0.436603i	$0.856181\pi$
$180$		0			0
$181$	−1.87674e29			−0.838653			−0.419327	−	0.907835i	$-0.637734\pi$
$181$	−1.87674e29			−0.838653			−0.419327	+	0.907835i	$0.637734\pi$
$182$		0			0
$183$	−1.34048e29	+	3.58779e28i	−0.519275	+	0.138983i
$184$		0			0
$185$		−	3.14822e28i		−	0.105884i
$186$		0			0
$187$	1.40835e27			0.00411880
$188$		0			0
$189$	−1.87174e29	+	1.86859e29i	−0.476696	+	0.475894i
$190$		0			0
$191$		−	2.40262e29i		−	0.533644i	−0.963746	−	0.266822i	$-0.914026\pi$
$191$		−	2.40262e29i		−	0.533644i	0.963746	−	0.266822i	$-0.0859737\pi$
$192$		0			0
$193$	−8.59177e29			−1.66662			−0.833312	−	0.552803i	$-0.813558\pi$
$193$	−8.59177e29			−1.66662			−0.833312	+	0.552803i	$0.813558\pi$
$194$		0			0
$195$	−1.88155e29	−	7.02994e29i	−0.319203	−	1.19262i
$196$		0			0
$197$		−	1.82524e29i		−	0.271181i	−0.990765	−	0.135591i	$-0.956707\pi$
$197$		−	1.82524e29i		−	0.271181i	0.990765	−	0.135591i	$-0.0432932\pi$
$198$		0			0
$199$	−6.18385e29			−0.805692			−0.402846	−	0.915268i	$-0.631979\pi$
$199$	−6.18385e29			−0.805692			−0.402846	+	0.915268i	$0.631979\pi$
$200$		0			0
$201$	6.36534e29	−	1.70368e29i	0.728238	−	0.194912i
$202$		0			0
$203$		−	9.70524e29i		−	0.976243i
$204$		0			0
$205$	−1.81670e30			−1.60873
$206$		0			0
$207$	5.19961e29	+	9.01766e29i	0.405841	+	0.703849i
$208$		0			0
$209$		−	3.31094e30i		−	2.28060i
$210$		0			0
$211$	3.84159e25			2.33797e−5			1.16899e−5	−	1.00000i	$-0.499996\pi$
$211$	3.84159e25			2.33797e−5			1.16899e−5		1.00000i	$0.499996\pi$
$212$		0			0
$213$	8.04161e29	+	3.00454e30i	0.432920	+	1.61750i
$214$		0			0
$215$			1.48362e30i			0.707330i
$216$		0			0
$217$	−6.34223e29			−0.268081
$218$		0			0
$219$	−3.95239e30	+	1.05785e30i	−1.48281	+	0.396873i
$220$		0			0
$221$		−	1.10454e28i		−	0.00368198i
$222$		0			0
$223$	−1.71022e30			−0.507096			−0.253548	−	0.967323i	$-0.581598\pi$
$223$	−1.71022e30			−0.507096			−0.253548	+	0.967323i	$0.581598\pi$
$224$		0			0
$225$	−6.49904e28	+	3.74737e28i	−0.0171584	+	0.00989358i
$226$		0			0
$227$		−	6.90133e29i		−	0.162404i	−0.996698	−	0.0812020i	$-0.974124\pi$
$227$		−	6.90133e29i		−	0.162404i	0.996698	−	0.0812020i	$-0.0258759\pi$
$228$		0			0
$229$	3.56889e30			0.749328			0.374664	−	0.927161i	$-0.377758\pi$
$229$	3.56889e30			0.749328			0.374664	+	0.927161i	$0.377758\pi$
$230$		0			0
$231$	−1.27012e30	−	4.74547e30i	−0.238171	−	0.889863i
$232$		0			0
$233$			4.74026e30i			0.794649i	0.917678	+	0.397325i	$0.130061\pi$
$233$			4.74026e30i			0.794649i	−0.917678	+	0.397325i	$0.869939\pi$
$234$		0			0
$235$	2.48608e29			0.0372934
$236$		0			0
$237$	−1.10295e31	+	2.95202e30i	−1.48192	+	0.396635i
$238$		0			0
$239$		−	7.68710e30i		−	0.925955i	−0.886370	−	0.462978i	$-0.846781\pi$
$239$		−	7.68710e30i		−	0.925955i	0.886370	−	0.462978i	$-0.153219\pi$
$240$		0			0
$241$	−1.02374e31			−1.10654			−0.553270	−	0.833002i	$-0.686620\pi$
$241$	−1.02374e31			−1.10654			−0.553270	+	0.833002i	$0.686620\pi$
$242$		0			0
$243$	−2.68006e30	+	9.94630e30i	−0.260173	+	0.965562i
$244$		0			0
$245$		−	6.32175e30i		−	0.551669i
$246$		0			0
$247$	−2.59669e31			−2.03873
$248$		0			0
$249$	−5.26438e30	−	1.96690e31i	−0.372182	−	1.39056i
$250$		0			0
$251$			3.66724e30i			0.233658i	0.993152	+	0.116829i	$0.0372729\pi$
$251$			3.66724e30i			0.233658i	−0.993152	+	0.116829i	$0.962727\pi$
$252$		0			0
$253$	−1.93344e31			−1.11113
$254$		0			0
$255$	−5.66334e28	+	1.51578e28i	−0.00293799	+	0.000786348i
$256$		0			0
$257$			6.75971e30i			0.316809i	0.987374	+	0.158405i	$0.0506350\pi$
$257$			6.75971e30i			0.316809i	−0.987374	+	0.158405i	$0.949365\pi$
$258$		0			0
$259$	−1.66672e30			−0.0706260
$260$		0			0
$261$	−1.88817e31	−	3.27464e31i	−0.723959	−	1.25556i
$262$		0			0
$263$		−	3.46577e31i		−	1.20330i	−0.798758	−	0.601652i	$-0.794509\pi$
$263$		−	3.46577e31i		−	1.20330i	0.798758	−	0.601652i	$-0.205491\pi$
$264$		0			0
$265$	−2.15280e31			−0.677341
$266$		0			0
$267$	−8.74107e30	−	3.26588e31i	−0.249413	−	0.931866i
$268$		0			0
$269$		−	3.16346e31i		−	0.819187i	−0.912268	−	0.409593i	$-0.865671\pi$
$269$		−	3.16346e31i		−	0.819187i	0.912268	−	0.409593i	$-0.134329\pi$
$270$		0			0
$271$	−7.38389e31			−1.73654			−0.868269	−	0.496093i	$-0.834767\pi$
$271$	−7.38389e31			−1.73654			−0.868269	+	0.496093i	$0.834767\pi$
$272$		0			0
$273$	−3.72176e31	+	9.96124e30i	−0.795489	+	0.212911i
$274$		0			0
$275$		−	1.39343e30i		−	0.0270870i
$276$		0			0
$277$	−5.06025e31			−0.895230			−0.447615	−	0.894226i	$-0.647727\pi$
$277$	−5.06025e31			−0.895230			−0.447615	+	0.894226i	$0.647727\pi$
$278$		0			0
$279$	−2.13993e31	+	1.23389e31i	−0.344783	+	0.198803i
$280$		0			0
$281$			9.60687e31i			1.41059i	0.708916	+	0.705293i	$0.249185\pi$
$281$			9.60687e31i			1.41059i	−0.708916	+	0.705293i	$0.750815\pi$
$282$		0			0
$283$	−2.44285e31			−0.327095			−0.163548	−	0.986535i	$-0.552294\pi$
$283$	−2.44285e31			−0.327095			−0.163548	+	0.986535i	$0.552294\pi$
$284$		0			0
$285$	3.56350e31	+	1.33141e32i	0.435405	+	1.62678i
$286$		0			0
$287$			9.61787e31i			1.07304i
$288$		0			0
$289$	9.80998e31			0.999991
$290$		0			0
$291$	9.51344e31	−	2.54626e31i	0.886602	−	0.237298i
$292$		0			0
$293$		−	2.03701e32i		−	1.73666i	−0.495989	−	0.868329i	$-0.665195\pi$
$293$		−	2.03701e32i		−	1.73666i	0.495989	−	0.868329i	$-0.334805\pi$
$294$		0			0
$295$	−2.16119e32			−1.68658
$296$		0			0
$297$	−1.35179e32	−	1.35407e32i	−0.966217	−	0.967845i
$298$		0			0
$299$			1.51635e32i			0.993286i
$300$		0			0
$301$	7.85451e31			0.471797
$302$		0			0
$303$	−5.81817e31	−	2.17381e32i	−0.320651	−	1.19803i
$304$		0			0
$305$			1.07295e32i			0.542850i
$306$		0			0
$307$	1.82282e32			0.847118			0.423559	−	0.905869i	$-0.360781\pi$
$307$	1.82282e32			0.847118			0.423559	+	0.905869i	$0.360781\pi$
$308$		0			0
$309$	1.77860e32	−	4.76041e31i	0.759660	−	0.203322i
$310$		0			0
$311$		−	1.88339e32i		−	0.739701i	−0.929091	−	0.369851i	$-0.879409\pi$
$311$		−	1.88339e32i		−	0.739701i	0.929091	−	0.369851i	$-0.120591\pi$
$312$		0			0
$313$	−2.20811e32			−0.797892			−0.398946	−	0.916974i	$-0.630624\pi$
$313$	−2.20811e32			−0.797892			−0.398946	+	0.916974i	$0.630624\pi$
$314$		0			0
$315$	1.02149e32	+	1.77157e32i	0.339780	+	0.589280i
$316$		0			0
$317$			1.52978e32i			0.468660i	0.972157	+	0.234330i	$0.0752897\pi$
$317$			1.52978e32i			0.468660i	−0.972157	+	0.234330i	$0.924710\pi$
$318$		0			0
$319$	7.02103e32			1.98208
$320$		0			0
$321$	3.10037e31	+	1.15837e32i	0.0806953	+	0.301497i
$322$		0			0
$323$			2.09190e30i			0.00502237i
$324$		0			0
$325$	−1.09284e31			−0.0242143
$326$		0			0
$327$	−2.71907e32	+	7.27755e31i	−0.556287	+	0.148890i
$328$		0			0
$329$		−	1.31617e31i		−	0.0248751i
$330$		0			0
$331$	−7.16363e31			−0.125132			−0.0625661	−	0.998041i	$-0.519928\pi$
$331$	−7.16363e31			−0.125132			−0.0625661	+	0.998041i	$0.519928\pi$
$332$		0			0
$333$	−5.62367e31	+	3.24262e31i	−0.0908332	+	0.0523747i
$334$		0			0
$335$		−	5.09493e32i		−	0.761299i
$336$		0			0
$337$	1.68287e32			0.232736			0.116368	−	0.993206i	$-0.462875\pi$
$337$	1.68287e32			0.232736			0.116368	+	0.993206i	$0.462875\pi$
$338$		0			0
$339$	2.23618e32	+	8.35492e32i	0.286359	+	1.06991i
$340$		0			0
$341$		−	4.58814e32i		−	0.544289i
$342$		0			0
$343$	−9.47337e32			−1.04155
$344$		0			0
$345$	7.77484e32	−	2.08093e32i	0.792581	−	0.212133i
$346$		0			0
$347$		−	1.44656e32i		−	0.136790i	−0.997658	−	0.0683948i	$-0.978212\pi$
$347$		−	1.44656e32i		−	0.136790i	0.997658	−	0.0683948i	$-0.0217878\pi$
$348$		0			0
$349$	−5.46703e32			−0.479757			−0.239878	−	0.970803i	$-0.577108\pi$
$349$	−5.46703e32			−0.479757			−0.239878	+	0.970803i	$0.577108\pi$
$350$		0			0
$351$	−1.06196e33	+	1.06018e33i	−0.865200	+	0.863745i
$352$		0			0
$353$		−	1.31891e31i		−	0.00998029i	−0.999988	−	0.00499015i	$-0.998412\pi$
$353$		−	1.31891e31i		−	0.00998029i	0.999988	−	0.00499015i	$-0.00158842\pi$
$354$		0			0
$355$	2.40488e33			1.69093
$356$		0			0
$357$	8.02480e29	+	2.99826e30i	0.000524503	+	0.00195967i
$358$		0			0
$359$		−	1.00670e33i		−	0.611888i	−0.952049	−	0.305944i	$-0.901028\pi$
$359$		−	1.00670e33i		−	0.611888i	0.952049	−	0.305944i	$-0.0989721\pi$
$360$		0			0
$361$	3.14946e33			1.78091
$362$		0			0
$363$	1.59746e33	−	4.27559e32i	0.840708	−	0.225014i
$364$		0			0
$365$			3.16356e33i			1.55013i
$366$		0			0
$367$	1.54039e33			0.703029			0.351514	−	0.936182i	$-0.385667\pi$
$367$	1.54039e33			0.703029			0.351514	+	0.936182i	$0.385667\pi$
$368$		0			0
$369$	1.87117e33	+	3.24517e33i	0.795739	+	1.38005i
$370$		0			0
$371$			1.13973e33i			0.451793i
$372$		0			0
$373$	2.89395e33			1.06973			0.534867	−	0.844936i	$-0.320362\pi$
$373$	2.89395e33			1.06973			0.534867	+	0.844936i	$0.320362\pi$
$374$		0			0
$375$	−7.42196e32	−	2.77302e33i	−0.255925	−	0.956196i
$376$		0			0
$377$		−	5.50643e33i		−	1.77187i
$378$		0			0
$379$	4.99203e33			1.49957			0.749786	−	0.661680i	$-0.230156\pi$
$379$	4.99203e33			1.49957			0.749786	+	0.661680i	$0.230156\pi$
$380$		0			0
$381$	2.52875e33	−	6.76815e32i	0.709383	−	0.189865i
$382$		0			0
$383$			4.67193e33i			1.22437i	0.790715	+	0.612184i	$0.209709\pi$
$383$			4.67193e33i			1.22437i	−0.790715	+	0.612184i	$0.790291\pi$
$384$		0			0
$385$	−3.79835e33			−0.930263
$386$		0			0
$387$	2.65019e33	−	1.52811e33i	0.606785	−	0.349874i
$388$		0			0
$389$			1.99238e33i			0.426605i	0.976986	+	0.213303i	$0.0684220\pi$
$389$			1.99238e33i			0.426605i	−0.976986	+	0.213303i	$0.931578\pi$
$390$		0			0
$391$	1.22158e31			0.00244694
$392$		0			0
$393$	−1.47187e33	−	5.49927e33i	−0.275909	−	1.03086i
$394$		0			0
$395$			8.82816e33i			1.54920i
$396$		0			0
$397$	−1.32621e33			−0.217939			−0.108969	−	0.994045i	$-0.534755\pi$
$397$	−1.32621e33			−0.217939			−0.108969	+	0.994045i	$0.534755\pi$
$398$		0			0
$399$	7.04870e33	−	1.88657e33i	1.08508	−	0.290420i
$400$		0			0
$401$			3.44450e33i			0.496878i	0.968648	+	0.248439i	$0.0799175\pi$
$401$			3.44450e33i			0.496878i	−0.968648	+	0.248439i	$0.920083\pi$
$402$		0			0
$403$	−3.59837e33			−0.486564
$404$		0			0
$405$	6.89324e33	+	3.99013e33i	0.873993	+	0.505908i
$406$		0			0
$407$		−	1.20575e33i		−	0.143393i
$408$		0			0
$409$	8.30731e33			0.926952			0.463476	−	0.886109i	$-0.346602\pi$
$409$	8.30731e33			0.926952			0.463476	+	0.886109i	$0.346602\pi$
$410$		0			0
$411$	1.11290e33	+	4.15805e33i	0.116550	+	0.435458i
$412$		0			0
$413$			1.14417e34i			1.12497i
$414$		0			0
$415$	−1.57434e34			−1.45370
$416$		0			0
$417$	−1.77569e34	+	4.75260e33i	−1.54027	+	0.412252i
$418$		0			0
$419$		−	1.64797e34i		−	1.34328i	−0.740879	−	0.671639i	$-0.765591\pi$
$419$		−	1.64797e34i		−	1.34328i	0.740879	−	0.671639i	$-0.234409\pi$
$420$		0			0
$421$	−7.23393e33			−0.554252			−0.277126	−	0.960834i	$-0.589382\pi$
$421$	−7.23393e33			−0.554252			−0.277126	+	0.960834i	$0.589382\pi$
$422$		0			0
$423$	−2.56062e32	−	4.44088e32i	−0.0184468	−	0.0319922i
$424$		0			0
$425$			8.80393e29i			5.96514e-5i
$426$		0			0
$427$	5.68034e33			0.362086
$428$		0			0
$429$	−7.20623e33	−	2.69242e34i	−0.432278	−	1.61510i
$430$		0			0
$431$		−	1.32435e34i		−	0.747821i	−0.927465	−	0.373911i	$-0.878017\pi$
$431$		−	1.32435e34i		−	0.747821i	0.927465	−	0.373911i	$-0.121983\pi$
$432$		0			0
$433$	2.51472e34			1.33705			0.668524	−	0.743691i	$-0.266926\pi$
$433$	2.51472e34			1.33705			0.668524	+	0.743691i	$0.266926\pi$
$434$		0			0
$435$	−2.82333e34	+	7.55661e33i	−1.41384	+	0.378413i
$436$		0			0
$437$		−	2.87184e34i		−	1.35488i
$438$		0			0
$439$	−3.90176e33			−0.173469			−0.0867344	−	0.996231i	$-0.527643\pi$
$439$	−3.90176e33			−0.173469			−0.0867344	+	0.996231i	$0.527643\pi$
$440$		0			0
$441$	−1.12926e34	+	6.51132e33i	−0.473250	+	0.272877i
$442$		0			0
$443$		−	1.27696e34i		−	0.504579i	−0.967652	−	0.252290i	$-0.918816\pi$
$443$		−	1.27696e34i		−	0.504579i	0.967652	−	0.252290i	$-0.0811836\pi$
$444$		0			0
$445$	−2.61406e34			−0.974172
$446$		0			0
$447$	7.32864e32	+	2.73816e33i	0.0257647	+	0.0962633i
$448$		0			0
$449$			8.73088e32i			0.0289638i	0.999895	+	0.0144819i	$0.00460990\pi$
$449$			8.73088e32i			0.0289638i	−0.999895	+	0.0144819i	$0.995390\pi$
$450$		0			0
$451$	−6.95783e34			−2.17861
$452$		0			0
$453$	−3.64484e32	+	9.75536e31i	−0.0107746	+	0.00288381i
$454$		0			0
$455$			2.97896e34i			0.831604i
$456$		0			0
$457$	−5.37341e34			−1.41690			−0.708450	−	0.705761i	$-0.750605\pi$
$457$	−5.37341e34			−1.41690			−0.708450	+	0.705761i	$0.750605\pi$
$458$		0			0
$459$	8.54081e31	+	8.55519e31i	0.00212782	+	0.00213140i
$460$		0			0
$461$		−	9.06898e33i		−	0.213524i	−0.994285	−	0.106762i	$-0.965952\pi$
$461$		−	9.06898e33i		−	0.213524i	0.994285	−	0.106762i	$-0.0340483\pi$
$462$		0			0
$463$	4.75355e34			1.05795			0.528976	−	0.848637i	$-0.322576\pi$
$463$	4.75355e34			1.05795			0.528976	+	0.848637i	$0.322576\pi$
$464$		0			0
$465$	4.93813e33	+	1.84501e34i	0.103914	+	0.388248i
$466$		0			0
$467$		−	4.90538e34i		−	0.976232i	−0.872779	−	0.488116i	$-0.837684\pi$
$467$		−	4.90538e34i		−	0.976232i	0.872779	−	0.488116i	$-0.162316\pi$
$468$		0			0
$469$	−2.69734e34			−0.507795
$470$		0			0
$471$	6.91074e34	−	1.84965e34i	1.23099	−	0.329472i
$472$		0			0
$473$			5.68216e34i			0.957897i
$474$		0			0
$475$	2.06974e33			0.0330293
$476$		0			0
$477$	2.21735e34	+	3.84555e34i	0.335039	+	0.581058i
$478$		0			0
$479$			1.03793e35i			1.48527i	0.669698	+	0.742633i	$0.266423\pi$
$479$			1.03793e35i			1.48527i	−0.669698	+	0.742633i	$0.733577\pi$
$480$		0			0
$481$	−9.45640e33			−0.128186
$482$		0			0
$483$	−1.10168e34	−	4.11613e34i	−0.141495	−	0.528659i
$484$		0			0
$485$		−	7.61471e34i		−	0.926853i
$486$		0			0
$487$	−7.44999e34			−0.859565			−0.429783	−	0.902932i	$-0.641410\pi$
$487$	−7.44999e34			−0.859565			−0.429783	+	0.902932i	$0.641410\pi$
$488$		0			0
$489$	−3.73057e34	+	9.98481e33i	−0.408093	+	0.109226i
$490$		0			0
$491$			2.15002e34i			0.223040i	0.993762	+	0.111520i	$0.0355719\pi$
$491$			2.15002e34i			0.223040i	−0.993762	+	0.111520i	$0.964428\pi$
$492$		0			0
$493$	−4.43600e32			−0.00436497
$494$		0			0
$495$	−1.28160e35	+	7.38975e34i	−1.19643	+	0.689862i
$496$		0			0
$497$		−	1.27318e35i		−	1.12787i
$498$		0			0
$499$	−1.00391e35			−0.844089			−0.422045	−	0.906575i	$-0.638687\pi$
$499$	−1.00391e35			−0.844089			−0.422045	+	0.906575i	$0.638687\pi$
$500$		0			0
$501$	2.71757e34	+	1.01535e35i	0.216915	+	0.810449i
$502$		0			0
$503$		−	1.24283e35i		−	0.941948i	−0.882147	−	0.470974i	$-0.843903\pi$
$503$		−	1.24283e35i		−	0.941948i	0.882147	−	0.470974i	$-0.156097\pi$
$504$		0			0
$505$	−1.73995e35			−1.25242
$506$		0			0
$507$	−6.98806e34	+	1.87034e34i	−0.477808	+	0.127885i
$508$		0			0
$509$		−	6.87195e34i		−	0.446425i	−0.974770	−	0.223213i	$-0.928346\pi$
$509$		−	6.87195e34i		−	0.446425i	0.974770	−	0.223213i	$-0.0716544\pi$
$510$		0			0
$511$	1.67484e35			1.03395
$512$		0			0
$513$	2.01127e35	−	2.00788e35i	1.18017	−	1.17818i
$514$		0			0
$515$		−	1.42363e35i		−	0.794148i
$516$		0			0
$517$	9.52151e33			0.0505043
$518$		0			0
$519$	−6.69528e34	−	2.50152e35i	−0.337748	−	1.26191i
$520$		0			0
$521$			1.10872e35i			0.532024i	0.963970	+	0.266012i	$0.0857061\pi$
$521$			1.10872e35i			0.532024i	−0.963970	+	0.266012i	$0.914294\pi$
$522$		0			0
$523$	−8.28471e34			−0.378229			−0.189115	−	0.981955i	$-0.560562\pi$
$523$	−8.28471e34			−0.378229			−0.189115	+	0.981955i	$0.560562\pi$
$524$		0			0
$525$	2.96650e33	−	7.93979e32i	0.0128876	−	0.00344936i
$526$		0			0
$527$			2.89886e32i			0.00119864i
$528$		0			0
$529$	8.63499e34			0.339890
$530$		0			0
$531$	2.22599e35	+	3.86054e35i	0.834248	+	1.44683i
$532$		0			0
$533$			5.45686e35i			1.94755i
$534$		0			0
$535$	9.27182e34			0.315185
$536$		0			0
$537$	−6.97146e34	−	2.60471e35i	−0.225766	−	0.843516i
$538$		0			0
$539$		−	2.42119e35i		−	0.747094i
$540$		0			0
$541$	−3.35003e35			−0.985108			−0.492554	−	0.870282i	$-0.663937\pi$
$541$	−3.35003e35			−0.985108			−0.492554	+	0.870282i	$0.663937\pi$
$542$		0			0
$543$	2.89039e35	−	7.73609e34i	0.810138	−	0.216832i
$544$		0			0
$545$			2.17639e35i			0.581543i
$546$		0			0
$547$	−1.04505e35			−0.266257			−0.133129	−	0.991099i	$-0.542502\pi$
$547$	−1.04505e35			−0.266257			−0.133129	+	0.991099i	$0.542502\pi$
$548$		0			0
$549$	1.91660e35	−	1.10512e35i	0.465685	−	0.268515i
$550$		0			0
$551$			1.04287e36i			2.41691i
$552$		0			0
$553$	4.67377e35			1.03333
$554$		0			0
$555$	1.29772e34	+	4.84861e34i	0.0273762	+	0.102284i
$556$		0			0
$557$		−	4.53529e35i		−	0.913035i	−0.889714	−	0.456518i	$-0.849097\pi$
$557$		−	4.53529e35i		−	0.913035i	0.889714	−	0.456518i	$-0.150903\pi$
$558$		0			0
$559$	4.45639e35			0.856307
$560$		0			0
$561$	−2.16902e33	+	5.80536e32i	−0.00397875	+	0.00106491i
$562$		0			0
$563$			2.28384e35i			0.399997i	0.979796	+	0.199998i	$0.0640937\pi$
$563$			2.28384e35i			0.399997i	−0.979796	+	0.199998i	$0.935906\pi$
$564$		0			0
$565$	6.68742e35			1.11848
$566$		0			0
$567$	2.11244e35	−	3.64939e35i	0.337446	−	0.582962i
$568$		0			0
$569$		−	8.90482e33i		−	0.0135883i	−0.999977	−	0.00679416i	$-0.997837\pi$
$569$		−	8.90482e33i		−	0.0135883i	0.999977	−	0.00679416i	$-0.00216266\pi$
$570$		0			0
$571$	8.88319e35			1.29509			0.647545	−	0.762028i	$-0.275796\pi$
$571$	8.88319e35			1.29509			0.647545	+	0.762028i	$0.275796\pi$
$572$		0			0
$573$	9.90384e34	+	3.70031e35i	0.137973	+	0.515500i
$574$		0			0
$575$		−	1.20864e34i		−	0.0160921i
$576$		0			0
$577$	3.80265e35			0.483950			0.241975	−	0.970282i	$-0.422205\pi$
$577$	3.80265e35			0.483950			0.241975	+	0.970282i	$0.422205\pi$
$578$		0			0
$579$	1.32323e36	−	3.54161e35i	1.60996	−	0.430903i
$580$		0			0
$581$			8.33481e35i			0.969630i
$582$		0			0
$583$	−8.24508e35			−0.917284
$584$		0			0
$585$	5.79561e35	+	1.00513e36i	0.616698	+	1.06954i
$586$		0			0
$587$		−	2.03170e35i		−	0.206806i	−0.994640	−	0.103403i	$-0.967027\pi$
$587$		−	2.03170e35i		−	0.206806i	0.994640	−	0.103403i	$-0.0329732\pi$
$588$		0			0
$589$	6.81501e35			0.663694
$590$		0			0
$591$	7.52382e34	+	2.81108e35i	0.0701134	+	0.261961i
$592$		0			0
$593$		−	8.83883e35i		−	0.788286i	−0.919049	−	0.394143i	$-0.871042\pi$
$593$		−	8.83883e35i		−	0.788286i	0.919049	−	0.394143i	$-0.128958\pi$
$594$		0			0
$595$	2.39986e33			0.00204864
$596$		0			0
$597$	9.52384e35	−	2.54904e35i	0.778297	−	0.208310i
$598$		0			0
$599$		−	1.00161e36i		−	0.783700i	−0.920029	−	0.391850i	$-0.871835\pi$
$599$		−	1.00161e36i		−	0.783700i	0.920029	−	0.391850i	$-0.128165\pi$
$600$		0			0
$601$	−6.62507e35			−0.496389			−0.248195	−	0.968710i	$-0.579837\pi$
$601$	−6.62507e35			−0.496389			−0.248195	+	0.968710i	$0.579837\pi$
$602$		0			0
$603$	−9.10108e35	+	5.24771e35i	−0.653082	+	0.376569i
$604$		0			0
$605$		−	1.27864e36i		−	0.878876i
$606$		0			0
$607$	−1.65391e36			−1.08908			−0.544539	−	0.838736i	$-0.683295\pi$
$607$	−1.65391e36			−1.08908			−0.544539	+	0.838736i	$0.683295\pi$
$608$		0			0
$609$	4.00059e35	+	1.49472e36i	0.252406	+	0.943049i
$610$		0			0
$611$		−	7.46750e34i		−	0.0451481i
$612$		0			0
$613$	−1.51404e35			−0.0877305			−0.0438653	−	0.999037i	$-0.513967\pi$
$613$	−1.51404e35			−0.0877305			−0.0438653	+	0.999037i	$0.513967\pi$
$614$		0			0
$615$	2.79792e36	−	7.48858e35i	1.55403	−	0.415933i
$616$		0			0
$617$		−	1.42196e36i		−	0.757145i	−0.925572	−	0.378573i	$-0.876415\pi$
$617$		−	1.42196e36i		−	0.757145i	0.925572	−	0.378573i	$-0.123585\pi$
$618$		0			0
$619$	−1.10493e36			−0.564102			−0.282051	−	0.959399i	$-0.591015\pi$
$619$	−1.10493e36			−0.564102			−0.282051	+	0.959399i	$0.591015\pi$
$620$		0			0
$621$	−1.17251e36	−	1.17449e36i	−0.574021	−	0.574987i
$622$		0			0
$623$			1.38393e36i			0.649783i
$624$		0			0
$625$	−2.26355e36			−1.01941
$626$		0			0
$627$	1.36480e36	+	5.09922e36i	0.589645	+	2.20306i
$628$		0			0
$629$			7.61810e32i			0.000315783i
$630$		0			0
$631$	−2.72500e36			−1.08389			−0.541945	−	0.840414i	$-0.682312\pi$
$631$	−2.72500e36			−1.08389			−0.541945	+	0.840414i	$0.682312\pi$
$632$		0			0
$633$	−5.91649e31	+	1.58354e31i	−2.25848e−5	+	6.04478e-6i
$634$		0			0
$635$		−	2.02405e36i		−	0.741589i
$636$		0			0
$637$	−1.89888e36			−0.667860
$638$		0			0
$639$	−2.47700e36	−	4.29585e36i	−0.836401	−	1.45057i
$640$		0			0
$641$			4.23037e36i			1.37159i	0.727795	+	0.685794i	$0.240545\pi$
$641$			4.23037e36i			1.37159i	−0.727795	+	0.685794i	$0.759455\pi$
$642$		0			0
$643$	4.42616e35			0.137811			0.0689055	−	0.997623i	$-0.478049\pi$
$643$	4.42616e35			0.137811			0.0689055	+	0.997623i	$0.478049\pi$
$644$		0			0
$645$	−6.11561e35	−	2.28494e36i	−0.182879	−	0.683280i
$646$		0			0
$647$		−	3.01210e36i		−	0.865196i	−0.901587	−	0.432598i	$-0.857597\pi$
$647$		−	3.01210e36i		−	0.865196i	0.901587	−	0.432598i	$-0.142403\pi$
$648$		0			0
$649$	−8.27721e36			−2.28404
$650$		0			0
$651$	9.76776e35	−	2.61433e35i	0.258966	−	0.0693118i
$652$		0			0
$653$			5.44100e36i			1.38614i	0.720870	+	0.693070i	$0.243742\pi$
$653$			5.44100e36i			1.38614i	−0.720870	+	0.693070i	$0.756258\pi$
$654$		0			0
$655$	−4.40170e36			−1.07767
$656$		0			0
$657$	5.65107e36	−	3.25842e36i	1.32978	−	0.766757i
$658$		0			0
$659$			7.89289e35i			0.178536i	0.996008	+	0.0892679i	$0.0284527\pi$
$659$			7.89289e35i			0.178536i	−0.996008	+	0.0892679i	$0.971547\pi$
$660$		0			0
$661$	1.16185e36			0.252658			0.126329	−	0.991988i	$-0.459681\pi$
$661$	1.16185e36			0.252658			0.126329	+	0.991988i	$0.459681\pi$
$662$		0			0
$663$	4.55301e33	+	1.70111e34i	0.000951968	+	0.00355678i
$664$		0			0
$665$		−	5.64190e36i		−	1.13434i
$666$		0			0
$667$	6.08990e36			1.17754
$668$		0			0
$669$	2.63394e36	−	7.04970e35i	0.489854	−	0.131109i
$670$		0			0
$671$			4.10931e36i			0.735151i
$672$		0			0
$673$	−5.33483e36			−0.918174			−0.459087	−	0.888391i	$-0.651823\pi$
$673$	−5.33483e36			−0.918174			−0.459087	+	0.888391i	$0.651823\pi$
$674$		0			0
$675$	8.46457e34	−	8.45033e34i	0.0140170	−	0.0139934i
$676$		0			0
$677$		−	9.77752e36i		−	1.55803i	−0.627005	−	0.779015i	$-0.715720\pi$
$677$		−	9.77752e36i		−	1.55803i	0.627005	−	0.779015i	$-0.284280\pi$
$678$		0			0
$679$	−4.03135e36			−0.618221
$680$		0			0
$681$	2.84479e35	+	1.06288e36i	0.0419892	+	0.156882i
$682$		0			0
$683$		−	8.66850e36i		−	1.23162i	−0.787897	−	0.615808i	$-0.788830\pi$
$683$		−	8.66850e36i		−	1.23162i	0.787897	−	0.615808i	$-0.211170\pi$
$684$		0			0
$685$	3.32817e36			0.455228
$686$		0			0
$687$	−5.49650e36	+	1.47113e36i	−0.723850	+	0.193737i
$688$		0			0
$689$			6.46643e36i			0.820001i
$690$		0			0
$691$	−1.30702e37			−1.59613			−0.798064	−	0.602573i	$-0.794142\pi$
$691$	−1.30702e37			−1.59613			−0.798064	+	0.602573i	$0.794142\pi$
$692$		0			0
$693$	3.91225e36	+	6.78501e36i	0.460145	+	0.798028i
$694$		0			0
$695$			1.42129e37i			1.61020i
$696$		0			0
$697$	4.39606e34			0.00479775
$698$		0			0
$699$	−1.95398e36	−	7.30054e36i	−0.205455	−	0.767630i
$700$		0			0
$701$			5.40690e36i			0.547790i	0.961760	+	0.273895i	$0.0883121\pi$
$701$			5.40690e36i			0.547790i	−0.961760	+	0.273895i	$0.911688\pi$
$702$		0			0
$703$	1.79096e36			0.174851
$704$		0			0
$705$	−3.82884e35	+	1.02478e35i	−0.0360254	+	0.00964214i
$706$		0			0
$707$			9.21159e36i			0.835377i
$708$		0			0
$709$	−2.22348e37			−1.94371			−0.971856	−	0.235576i	$-0.924302\pi$
$709$	−2.22348e37			−1.94371			−0.971856	+	0.235576i	$0.924302\pi$
$710$		0			0
$711$	1.57698e37	−	9.09289e36i	1.32899	−	0.766297i
$712$		0			0
$713$		−	3.97966e36i		−	0.323357i
$714$		0			0
$715$	−2.15506e37			−1.68842
$716$		0			0
$717$	3.16869e36	+	1.18390e37i	0.239404	+	0.894471i
$718$		0			0
$719$		−	9.65459e36i		−	0.703490i	−0.936096	−	0.351745i	$-0.885588\pi$
$719$		−	9.65459e36i		−	0.703490i	0.936096	−	0.351745i	$-0.114412\pi$
$720$		0			0
$721$	−7.53690e36			−0.529705
$722$		0			0
$723$	1.57667e37	−	4.21994e36i	1.06892	−	0.286094i
$724$		0			0
$725$			4.38900e35i			0.0287059i
$726$		0			0
$727$	7.33417e36			0.462811			0.231406	−	0.972857i	$-0.425668\pi$
$727$	7.33417e36			0.462811			0.231406	+	0.972857i	$0.425668\pi$
$728$		0			0
$729$	2.76386e34	−	1.64232e37i	0.00168290	−	0.999999i
$730$		0			0
$731$		−	3.59008e34i		−	0.00210949i
$732$		0			0
$733$	−3.17191e37			−1.79874			−0.899372	−	0.437184i	$-0.855976\pi$
$733$	−3.17191e37			−1.79874			−0.899372	+	0.437184i	$0.855976\pi$
$734$		0			0
$735$	2.60588e36	+	9.73622e36i	0.142633	+	0.532911i
$736$		0			0
$737$		−	1.95132e37i		−	1.03098i
$738$		0			0
$739$	−2.99143e37			−1.52581			−0.762906	−	0.646509i	$-0.776228\pi$
$739$	−2.99143e37			−1.52581			−0.762906	+	0.646509i	$0.776228\pi$
$740$		0			0
$741$	3.99920e37	−	1.07038e37i	1.96941	−	0.527110i
$742$		0			0
$743$		−	2.54738e37i		−	1.21126i	−0.795745	−	0.605632i	$-0.792920\pi$
$743$		−	2.54738e37i		−	1.21126i	0.795745	−	0.605632i	$-0.207080\pi$
$744$		0			0
$745$	2.19167e36			0.100634
$746$		0			0
$747$	1.62155e37	+	2.81225e37i	0.719055	+	1.24706i
$748$		0			0
$749$		−	4.90865e36i		−	0.210232i
$750$		0			0
$751$	6.06599e36			0.250947			0.125474	−	0.992097i	$-0.459955\pi$
$751$	6.06599e36			0.250947			0.125474	+	0.992097i	$0.459955\pi$
$752$		0			0
$753$	−1.51167e36	−	5.64796e36i	−0.0604118	−	0.225713i
$754$		0			0
$755$			2.91739e35i			0.0112638i
$756$		0			0
$757$	4.44956e37			1.65986			0.829928	−	0.557870i	$-0.188381\pi$
$757$	4.44956e37			1.65986			0.829928	+	0.557870i	$0.188381\pi$
$758$		0			0
$759$	2.97772e37	−	7.96981e36i	1.07335	−	0.287280i
$760$		0			0
$761$			3.56186e37i			1.24073i	0.784314	+	0.620364i	$0.213015\pi$
$761$			3.56186e37i			1.24073i	−0.784314	+	0.620364i	$0.786985\pi$
$762$		0			0
$763$	1.15221e37			0.387895
$764$		0			0
$765$	8.09736e34	−	4.66896e34i	0.00263478	−	0.00151922i
$766$		0			0
$767$			6.49163e37i			2.04180i
$768$		0			0
$769$	−3.93126e37			−1.19533			−0.597667	−	0.801744i	$-0.703906\pi$
$769$	−3.93126e37			−1.19533			−0.597667	+	0.801744i	$0.703906\pi$
$770$		0			0
$771$	−2.78642e36	−	1.04107e37i	−0.0819104	−	0.306037i
$772$		0			0
$773$			5.09642e37i			1.44855i	0.689514	+	0.724273i	$0.257824\pi$
$773$			5.09642e37i			1.44855i	−0.689514	+	0.724273i	$0.742176\pi$
$774$		0			0
$775$	2.86815e35			0.00788279
$776$		0			0
$777$	2.56693e36	−	6.87036e35i	0.0682246	−	0.0182602i
$778$		0			0
$779$		−	1.03348e38i		−	2.65654i
$780$		0			0
$781$	9.21055e37			2.28993
$782$		0			0
$783$	4.25783e37	+	4.26500e37i	1.02397	+	1.02569i
$784$		0			0
$785$		−	5.53148e37i		−	1.28687i
$786$		0			0
$787$	−2.84046e37			−0.639319			−0.319660	−	0.947532i	$-0.603569\pi$
$787$	−2.84046e37			−0.639319			−0.319660	+	0.947532i	$0.603569\pi$
$788$		0			0
$789$	1.42862e37	+	5.33768e37i	0.311112	+	1.16239i
$790$		0			0
$791$		−	3.54043e37i		−	0.746038i
$792$		0			0
$793$	3.22284e37			0.657184
$794$		0			0
$795$	3.31555e37	−	8.87404e36i	0.654310	−	0.175125i
$796$		0			0
$797$		−	2.82373e37i		−	0.539343i	−0.962952	−	0.269671i	$-0.913085\pi$
$797$		−	2.82373e37i		−	0.539343i	0.962952	−	0.269671i	$-0.0869151\pi$
$798$		0			0
$799$	−6.01584e33			−0.000111221
$800$		0			0
$801$	2.69245e37	+	4.66950e37i	0.481864	+	0.835696i
$802$		0			0
$803$			1.21162e38i			2.09926i
$804$		0			0
$805$	−3.29462e37			−0.552661
$806$		0			0
$807$	1.30401e37	+	4.87209e37i	0.211799	+	0.791333i
$808$		0			0
$809$			1.01040e38i			1.58914i	0.607170	+	0.794572i	$0.292305\pi$
$809$			1.01040e38i			1.58914i	−0.607170	+	0.794572i	$0.707695\pi$
$810$		0			0
$811$	3.30637e37			0.503594			0.251797	−	0.967780i	$-0.418978\pi$
$811$	3.30637e37			0.503594			0.251797	+	0.967780i	$0.418978\pi$
$812$		0			0
$813$	1.13720e38	−	3.04371e37i	1.67749	−	0.448979i
$814$		0			0
$815$			2.98601e37i			0.426621i
$816$		0			0
$817$	−8.44002e37			−1.16804
$818$		0			0
$819$	5.32132e37	−	3.06829e37i	0.713393	−	0.411344i
$820$		0			0
$821$		−	1.25524e38i		−	1.63029i	−0.579255	−	0.815146i	$-0.696656\pi$
$821$		−	1.25524e38i		−	1.63029i	0.579255	−	0.815146i	$-0.303344\pi$
$822$		0			0
$823$	−5.67168e37			−0.713697			−0.356849	−	0.934162i	$-0.616149\pi$
$823$	−5.67168e37			−0.713697			−0.356849	+	0.934162i	$0.616149\pi$
$824$		0			0
$825$	5.74386e35	+	2.14605e36i	0.00700330	+	0.0261660i
$826$		0			0
$827$			9.79011e37i			1.15669i	0.815793	+	0.578343i	$0.196301\pi$
$827$			9.79011e37i			1.15669i	−0.815793	+	0.578343i	$0.803699\pi$
$828$		0			0
$829$	1.00335e37			0.114880			0.0574400	−	0.998349i	$-0.481706\pi$
$829$	1.00335e37			0.114880			0.0574400	+	0.998349i	$0.481706\pi$
$830$		0			0
$831$	7.79336e37	−	2.08588e37i	0.864791	−	0.231460i
$832$		0			0
$833$			1.52975e35i			0.00164526i
$834$		0			0
$835$	8.12704e37			0.847243
$836$		0			0
$837$	2.78712e37	−	2.78243e37i	0.281659	−	0.281186i
$838$		0			0
$839$		−	1.06504e38i		−	1.04342i	−0.853123	−	0.521710i	$-0.825294\pi$
$839$		−	1.06504e38i		−	1.04342i	0.853123	−	0.521710i	$-0.174706\pi$
$840$		0			0
$841$	−1.15866e38			−1.10055
$842$		0			0
$843$	−3.96004e37	−	1.47957e38i	−0.364705	−	1.36262i
$844$		0			0
$845$			5.59336e37i			0.499500i
$846$		0			0
$847$	−6.76931e37			−0.586219
$848$		0			0
$849$	3.76227e37	−	1.00697e37i	0.315973	−	0.0845698i
$850$		0			0
$851$		−	1.04584e37i		−	0.0851886i
$852$		0			0
$853$	−2.09525e38			−1.65538			−0.827690	−	0.561186i	$-0.810345\pi$
$853$	−2.09525e38			−1.65538			−0.827690	+	0.561186i	$0.810345\pi$
$854$		0			0
$855$	−1.09764e38	−	1.90363e38i	−0.841202	−	1.45889i
$856$		0			0
$857$		−	2.08673e38i		−	1.55137i	−0.631121	−	0.775684i	$-0.717405\pi$
$857$		−	2.08673e38i		−	1.55137i	0.631121	−	0.775684i	$-0.282595\pi$
$858$		0			0
$859$	−8.21778e37			−0.592711			−0.296356	−	0.955078i	$-0.595771\pi$
$859$	−8.21778e37			−0.592711			−0.296356	+	0.955078i	$0.595771\pi$
$860$		0			0
$861$	−3.96458e37	−	1.48126e38i	−0.277432	−	1.03655i
$862$		0			0
$863$		−	5.83092e37i		−	0.395910i	−0.980211	−	0.197955i	$-0.936570\pi$
$863$		−	5.83092e37i		−	0.395910i	0.980211	−	0.197955i	$-0.0634300\pi$
$864$		0			0
$865$	−2.00226e38			−1.31920
$866$		0			0
$867$	−1.51085e38	+	4.04376e37i	−0.965990	+	0.258546i
$868$		0			0
$869$			3.38113e38i			2.09800i
$870$		0			0
$871$	−1.53038e38			−0.921643
$872$		0			0
$873$	−1.36022e38	+	7.84305e37i	−0.795103	+	0.458458i
$874$		0			0
$875$			1.17508e38i			0.666749i
$876$		0			0
$877$	2.86982e38			1.58074			0.790369	−	0.612632i	$-0.209889\pi$
$877$	2.86982e38			1.58074			0.790369	+	0.612632i	$0.209889\pi$
$878$		0			0
$879$	8.39674e37	+	3.13723e38i	0.449009	+	1.67761i
$880$		0			0
$881$			6.75270e37i			0.350583i	0.984517	+	0.175291i	$0.0560868\pi$
$881$			6.75270e37i			0.350583i	−0.984517	+	0.175291i	$0.943913\pi$
$882$		0			0
$883$	−7.62414e37			−0.384328			−0.192164	−	0.981363i	$-0.561550\pi$
$883$	−7.62414e37			−0.384328			−0.192164	+	0.981363i	$0.561550\pi$
$884$		0			0
$885$	3.32848e38	−	8.90862e37i	1.62923	−	0.436062i
$886$		0			0
$887$		−	1.38408e38i		−	0.657889i	−0.944349	−	0.328944i	$-0.893307\pi$
$887$		−	1.38408e38i		−	0.657889i	0.944349	−	0.328944i	$-0.106693\pi$
$888$		0			0
$889$	−1.07156e38			−0.494647
$890$		0			0
$891$	2.64007e38	+	1.52819e38i	1.18360	+	0.685123i
$892$		0			0
$893$			1.41428e37i			0.0615838i
$894$		0			0
$895$	−2.08485e38			−0.881812
$896$		0			0
$897$	−6.25054e37	−	2.33535e38i	−0.256812	−	0.959513i
$898$		0			0
$899$			1.44516e38i			0.576820i
$900$		0			0
$901$	5.20937e35			0.00202005
$902$		0			0
$903$	−1.20968e38	+	3.23770e37i	−0.455755	+	0.121982i
$904$		0			0
$905$		−	2.31352e38i		−	0.846918i
$906$		0			0
$907$	−2.62402e38			−0.933411			−0.466706	−	0.884413i	$-0.654559\pi$
$907$	−2.62402e38			−0.933411			−0.466706	+	0.884413i	$0.654559\pi$
$908$		0			0
$909$	1.79213e38	+	3.10808e38i	0.619496	+	1.07439i
$910$		0			0
$911$		−	1.63746e38i		−	0.550089i	−0.961432	−	0.275044i	$-0.911307\pi$
$911$		−	1.63746e38i		−	0.550089i	0.961432	−	0.275044i	$-0.0886925\pi$
$912$		0			0
$913$	−6.02962e38			−1.96866
$914$		0			0
$915$	−4.42278e37	−	1.65246e38i	−0.140353	−	0.524392i
$916$		0			0
$917$			2.33033e38i			0.718814i
$918$		0			0
$919$	2.98598e38			0.895334			0.447667	−	0.894200i	$-0.352255\pi$
$919$	2.98598e38			0.895334			0.447667	+	0.894200i	$0.352255\pi$
$920$		0			0
$921$	−2.80734e38	+	7.51381e37i	−0.818315	+	0.219021i
$922$		0			0
$923$		−	7.22362e38i		−	2.04707i
$924$		0			0
$925$	7.53740e35			0.00207673
$926$		0			0
$927$	−2.54302e38	+	1.46631e38i	−0.681262	+	0.392817i
$928$		0			0
$929$		−	4.73188e38i		−	1.23262i	−0.787503	−	0.616310i	$-0.788627\pi$
$929$		−	4.73188e38i		−	1.23262i	0.787503	−	0.616310i	$-0.211373\pi$
$930$		0			0
$931$	3.59633e38			0.910989
$932$		0			0
$933$	7.76352e37	+	2.90064e38i	0.191248	+	0.714550i
$934$		0			0
$935$			1.73612e36i			0.00415939i
$936$		0			0
$937$	−3.11039e37			−0.0724770			−0.0362385	−	0.999343i	$-0.511538\pi$
$937$	−3.11039e37			−0.0724770			−0.0362385	+	0.999343i	$0.511538\pi$
$938$		0			0
$939$	3.40074e38	−	9.10203e37i	0.770763	−	0.206294i
$940$		0			0
$941$		−	1.36991e37i		−	0.0302013i	−0.999886	−	0.0151007i	$-0.995193\pi$
$941$		−	1.36991e37i		−	0.0302013i	0.999886	−	0.0151007i	$-0.00480687\pi$
$942$		0			0
$943$	−6.03508e38			−1.29429
$944$		0			0
$945$	−2.30347e38	−	2.30735e38i	−0.480584	−	0.481394i
$946$		0			0
$947$			9.70144e38i			1.96919i	0.174864	+	0.984593i	$0.444052\pi$
$947$			9.70144e38i			1.96919i	−0.174864	+	0.984593i	$0.555948\pi$
$948$		0			0
$949$	9.50248e38			1.87662
$950$		0			0
$951$	−6.30589e37	−	2.35603e38i	−0.121171	−	0.452725i
$952$		0			0
$953$		−	1.80732e38i		−	0.337931i	−0.985622	−	0.168966i	$-0.945957\pi$
$953$		−	1.80732e38i		−	0.337931i	0.985622	−	0.168966i	$-0.0540427\pi$
$954$		0			0
$955$	2.96179e38			0.538903
$956$		0			0
$957$	−1.08132e39	+	2.89413e38i	−1.91469	+	0.512464i
$958$		0			0
$959$		−	1.76199e38i		−	0.303641i
$960$		0			0
$961$	−5.01777e38			−0.841602
$962$		0			0
$963$	−9.54984e37	−	1.65623e38i	−0.155903	−	0.270382i
$964$		0			0
$965$		−	1.05914e39i		−	1.68305i
$966$		0			0
$967$	5.18014e38			0.801304			0.400652	−	0.916230i	$-0.368784\pi$
$967$	5.18014e38			0.801304			0.400652	+	0.916230i	$0.368784\pi$
$968$		0			0
$969$	−8.62301e35	−	3.22177e36i	−0.00129852	−	0.00485160i
$970$		0			0
$971$			4.02923e38i			0.590706i	0.955388	+	0.295353i	$0.0954373\pi$
$971$			4.02923e38i			0.590706i	−0.955388	+	0.295353i	$0.904563\pi$
$972$		0			0
$973$	7.52453e38			1.07402
$974$		0			0
$975$	1.68309e37	−	4.50478e36i	0.0233910	−	0.00626056i
$976$		0			0
$977$			7.78610e38i			1.05364i	0.849978	+	0.526818i	$0.176615\pi$
$977$			7.78610e38i			1.05364i	−0.849978	+	0.526818i	$0.823385\pi$
$978$		0			0
$979$	−1.00117e39			−1.31927
$980$		0			0
$981$	3.88769e38	−	2.24165e38i	0.498878	−	0.287654i
$982$		0			0
$983$			5.24245e38i			0.655147i	0.944826	+	0.327574i	$0.106231\pi$
$983$			5.24245e38i			0.655147i	−0.944826	+	0.327574i	$0.893769\pi$
$984$		0			0
$985$	2.25004e38			0.273854
$986$		0			0
$987$	5.42537e36	+	2.02705e37i	0.00643141	+	0.0240293i
$988$		0			0
$989$			4.92860e38i			0.569078i
$990$		0			0
$991$	2.10773e37			0.0237059			0.0118530	−	0.999930i	$-0.496227\pi$
$991$	2.10773e37			0.0237059			0.0118530	+	0.999930i	$0.496227\pi$
$992$		0			0
$993$	1.10328e38	−	2.95291e37i	0.120877	−	0.0323527i
$994$		0			0
$995$		−	7.62304e38i		−	0.813632i
$996$		0			0
$997$	1.52641e39			1.58721			0.793604	−	0.608434i	$-0.208202\pi$
$997$	1.52641e39			1.58721			0.793604	+	0.608434i	$0.208202\pi$
$998$		0			0
$999$	7.32445e37	−	7.31213e37i	0.0742034	−	0.0740786i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.27.e.b.17.2		8
3.2	odd	2	inner	48.27.e.b.17.1		8
4.3	odd	2		6.27.b.a.5.4	✓	8
12.11	even	2		6.27.b.a.5.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6.27.b.a.5.4	✓	8	4.3	odd	2
6.27.b.a.5.8	yes	8	12.11	even	2
48.27.e.b.17.1		8	3.2	odd	2	inner
48.27.e.b.17.2		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 27 \)
Character orbit:	\([\chi]\)	\(=\)	48.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.54011e6 + 412209. i) q^{3} +1.23273e9i q^{5} +6.52628e10 q^{7} +(2.20203e12 - 1.26970e12i) q^{9} +O(q^{10})\)	\(q+(-1.54011e6 + 412209. i) q^{3} +1.23273e9i q^{5} +6.52628e10 q^{7} +(2.20203e12 - 1.26970e12i) q^{9} +4.72129e13i q^{11} +3.70280e14 q^{13} +(-5.08144e14 - 1.89855e15i) q^{15} -2.98298e13i q^{17} -7.01278e16 q^{19} +(-1.00512e17 + 2.69019e16i) q^{21} +4.09515e17i q^{23} -2.95138e16 q^{25} +(-2.86800e18 + 2.86318e18i) q^{27} -1.48710e19i q^{29} -9.71798e18 q^{31} +(-1.94616e19 - 7.27132e19i) q^{33} +8.04516e19i q^{35} -2.55385e19 q^{37} +(-5.70273e20 + 1.52633e20i) q^{39} +1.47371e21i q^{41} +1.20352e21 q^{43} +(1.56520e21 + 2.71452e21i) q^{45} -2.01672e20i q^{47} -5.12824e21 q^{49} +(1.22961e19 + 4.59413e19i) q^{51} +1.74636e22i q^{53} -5.82009e22 q^{55} +(1.08005e23 - 2.89073e22i) q^{57} +1.75317e23i q^{59} +8.70380e22 q^{61} +(1.43711e23 - 8.28641e22i) q^{63} +4.56456e23i q^{65} -4.13304e23 q^{67} +(-1.68806e23 - 6.30700e23i) q^{69} -1.95086e24i q^{71} +2.56630e24 q^{73} +(4.54547e22 - 1.21659e22i) q^{75} +3.08125e24i q^{77} +7.16146e24 q^{79} +(3.23682e24 - 5.59183e24i) q^{81} +1.27711e25i q^{83} +3.67722e22 q^{85} +(6.12996e24 + 2.29030e25i) q^{87} +2.12054e25i q^{89} +2.41655e25 q^{91} +(1.49668e25 - 4.00584e24i) q^{93} -8.64489e25i q^{95} -6.17710e25 q^{97} +(5.99461e25 + 1.03964e26i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2194920 q^{3} - 183211263760 q^{7} - 4853931303096 q^{9}+O(q^{10}) \) 8 * q - 2194920 * q^3 - 183211263760 * q^7 - 4853931303096 * q^9	\( 8 q - 2194920 q^{3} - 183211263760 q^{7} - 4853931303096 q^{9} + 512429022548560 q^{13} + 10\!\cdots\!40 q^{15}+ \cdots - 49\!\cdots\!36 q^{99}+O(q^{100}) \) 8 * q - 2194920 * q^3 - 183211263760 * q^7 - 4853931303096 * q^9 + 512429022548560 * q^13 + 1064426665559040 * q^15 - 13250502825732304 * q^19 - 376087240418977584 * q^21 - 5623831579932691000 * q^25 - 11144542948278241320 * q^27 + 31284409294003569776 * q^31 - 27135100230528798720 * q^33 - 884494012218278308400 * q^37 - 1550610890749922311440 * q^39 + 5161357724259210085040 * q^43 + 20490779480380306513920 * q^45 - 29984991896355791854440 * q^49 - 55593505357359326429184 * q^51 + 76225105088113564293120 * q^55 + 66535314651975639764880 * q^57 + 235221054408754640644432 * q^61 + 635461358798759743298160 * q^63 - 4218736573110688595120080 * q^67 - 2608639506957020943089664 * q^69 + 1648536069669962768198800 * q^73 - 2584159434345978668025000 * q^75 + 9378866636279258011040624 * q^79 - 746539638785265647226744 * q^81 - 41457721476781794937282560 * q^85 - 55271553720839002987729920 * q^87 + 71633324581325574226388320 * q^91 + 18549099857564620426167120 * q^93 - 92292347027073928712964080 * q^97 - 4908019641378975318951936 * q^99

Introduction

L-functions

Modular forms

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Database

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